scholarly journals A Characterization of the Existence of a Fundamental Bounded Resolution for the Space Cc(X) in Terms of X

2018 ◽  
Vol 2018 ◽  
pp. 1-5 ◽  
Author(s):  
Juan Carlos Ferrando
Keyword(s):  
Continuous Functions ◽  
Compact Open Topology ◽  
Tychonoff Space ◽  
Bounded Sets

We characterize in terms of the topology of a Tychonoff space X the existence of a bounded resolution for CcX that swallows the bounded sets, where CcX is the space of real-valued continuous functions on X equipped with the compact-open topology.

scholarly journals Convolution operators on weighted spaces of continuous functions and supremal convolution

2019 ◽  
Vol 199 (4) ◽  
pp. 1547-1569
Author(s):  
T. Kleiner ◽  
R. Hilfer
Keyword(s):  
Continuous Functions ◽  
Weighted Spaces ◽  
Constructive Method ◽  
Convolution Operators ◽  
Linear Combinations ◽  
The Third ◽  
Bounded Set ◽  
Spaces Of Continuous Functions ◽  
Bounded Sets

AbstractThe convolution of two weighted balls of measures is proved to be contained in a third weighted ball if and only if the supremal convolution of the corresponding two weights is less than or equal to the third weight. Here supremal convolution is introduced as a type of convolution in which integration is replaced with supremum formation. Invoking duality the equivalence implies a characterization of equicontinuity of weight-bounded sets of convolution operators having weighted spaces of continuous functions as domain and range. The overall result is a constructive method to define weighted spaces on which a given set of convolution operators acts as an equicontinuous family of endomorphisms. The result is applied to linear combinations of fractional Weyl integrals and derivatives with orders and coefficients from a given bounded set.

Natural extension of EF-spaces and EZ-spaces to the pointfree context

2019 ◽  
Vol 69 (5) ◽  
pp. 979-988
Author(s):  
Jissy Nsonde Nsayi
Keyword(s):  
Closed Subset ◽  
Natural Extension ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Tychonoff Space ◽  
Regular Closed

Abstract Two problems concerning EF-frames and EZ-frames are investigated. In [Some new classes of topological spaces and annihilator ideals, Topology Appl. 165 (2014), 84–97], Tahirefar defines a Tychonoff space X to be an EF (resp., EZ)-space if disjoint unions of clopen sets are completely separated (resp., every regular closed subset is the closure of a union of clopen subsets). By extending these notions to locales, we give several characterizations of EF and EZ-frames, mostly in terms of certain ring-theoretic properties of 𝓡 L, the ring of real-valued continuous functions on L. We end by defining a qsz-frame which is a pointfree context of qsz-space and, give a characterization of these frames in terms of rings of real-valued continuous functions on L.

Montel Algebras on the Plane

1970 ◽  
Vol 22 (1) ◽  
pp. 116-122 ◽  
Author(s):  
W. E. Meyers
Keyword(s):  
Topological Algebra ◽  
Maximum Modulus ◽  
Continuous Homomorphism ◽  
Continuous Functions ◽  
Maximum Modulus Principle ◽  
Compact Open Topology ◽  
Bounded Functions ◽  
Continuous Complex ◽  
Complex Valued

The results of Rudin in [7] show that under certain conditions, the maximum modulus principle characterizes the algebra A (G) of functions analytic on an open subset G of the plane C (see below). In [2], Birtel obtained a characterization of A(C) in terms of the Liouville theorem; he proved that every singly generated F-algebra of continuous functions on C which contains no non-constant bounded functions is isomorphic to A(C) in the compact-open topology. In this paper we show that the Montel property of the topological algebra A (G) also characterizes it. In particular, any Montel algebra A of continuous complex-valued functions on G which contains the polynomials and has continuous homomorphism space M (A) homeomorphic to G is precisely A(G).

scholarly journals Characterization of pseudocompactness by the topology of uniform convergence on function spaces

1978 ◽  
Vol 26 (2) ◽  
pp. 251-256 ◽  
Author(s):  
R. A. McCoy
Keyword(s):  
Uniform Convergence ◽  
Continuous Functions ◽  
Function Spaces ◽  
Metrizable Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Space Of Continuous Functions ◽  
Tychonoff Space ◽  
Necessary And Sufficient

AbstractIt is shown that a Tychonoff space X is pseudocompact if and only if for every metrizable space Y, all uniformities on Y induce the same topology on the space of continuous functions from X into Y. Also for certain pairs of spaces X and Y, a necessary and sufficient condition is established in order that all uniformities on Y induce the same topology on the space of continuous functions from X into Y.

scholarly journals Selection principles and games in bitopological function spaces

Filomat ◽  
2019 ◽  
Vol 33 (14) ◽  
pp. 4535-4540
Author(s):  
Daniil Lyakhovets ◽  
Alexander Osipov
Keyword(s):  
Pointwise Convergence ◽  
Continuous Functions ◽  
Function Spaces ◽  
Compact Open Topology ◽  
Bitopological Space ◽  
Tychonoff Space ◽  
Selection Principles ◽  
Topology Of Pointwise Convergence ◽  
Selective Separability

For a Tychonoff space X, we denote by (C(X), ?k ?p) the bitopological space of all real-valued continuous functions on X, where ?k is the compact-open topology and ?p is the topology of pointwise convergence. In the papers [6, 7, 13] variations of selective separability and tightness in (C(X),?k,?p) were investigated. In this paper we continue to study the selective properties and the corresponding topological games in the space (C(X),?k,?p).

scholarly journals Selection principles in function spaces with the compact-open topology

Filomat ◽  
2018 ◽  
Vol 32 (15) ◽  
pp. 5403-5413 ◽  
Author(s):  
Alexander Osipov
Keyword(s):  
Convergent Sequence ◽  
Continuous Functions ◽  
Function Spaces ◽  
Compact Open Topology ◽  
Tychonoff Space ◽  
Selection Principles

For a Tychonoff space X, we denote by Ck(X) the space of all real-valued continuous functions on X with the compact-open topology. A subset A ? X is said to be sequentially dense in X if every point of X is the limit of a convergent sequence in A. In this paper, the following properties for Ck(X) are considered. S1(S,S)=> Sfin(S,S) => Sfin(S,D) <=S1(S,D) S1(D,S) => Sfin(D,S) => Sfin(D,D) <= S1(D,D) For example, a space Ck(X) satisfies S1(S,D) (resp., Sfin(S,D)) if whenever (Sn : n ? N) is a sequence of sequentially dense subsets of Ck(X), one can take points fn ? Sn (resp., finite Fn ? Sn) such that {fn : n ? N} (resp.,U {Fn : n ? Ng) is dense in Ck(X). Other properties are defined similarly. In [22], we obtained characterizations these selection properties for Cp(X). In this paper, we give characterizations for Ck(X).

scholarly journals Homotopy Characterization of ANR Function Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Jaka Smrekar
Keyword(s):  
Topological Space ◽  
Homotopy Type ◽  
Metric Spaces ◽  
Function Spaces ◽  
Compact Open Topology ◽  
Tychonoff Space ◽  
Cw Complex ◽  
Continuous Maps ◽  
Compactly Generated

LetYbe an absolute neighbourhood retract (ANR) for the class of metric spaces and letXbe a topological space. LetYXdenote the space of continuous maps fromXtoYequipped with the compact open topology. We show that ifXis a compactly generated Tychonoff space andYis not discrete, thenYXis an ANR for metric spaces if and only ifXis hemicompact andYXhas the homotopy type of a CW complex.

scholarly journals Bounded resolutions for spaces $$C_{p}(X)$$ and a characterization in terms of X

2021 ◽  
Vol 115 (2) ◽  
Author(s):  
J. C. Ferrando ◽  
J. Ka̧kol ◽  
W. Śliwa
Keyword(s):  
Tychonoff Space ◽  
Cosmic Space ◽  
Tychonoff Spaces

AbstractAn internal characterization of the Arkhangel’skiĭ-Calbrix main theorem from [4] is obtained by showing that the space $$C_{p}(X)$$ C p ( X ) of continuous real-valued functions on a Tychonoff space X is K-analytic framed in $$\mathbb {R}^{X}$$ R X if and only if X admits a nice framing. This applies to show that a metrizable (or cosmic) space X is $$\sigma $$ σ -compact if and only if X has a nice framing. We analyse a few concepts which are useful while studying nice framings. For example, a class of Tychonoff spaces X containing strictly Lindelöf Čech-complete spaces is introduced for which a variant of Arkhangel’skiĭ-Calbrix theorem for $$\sigma $$ σ -boundedness of X is shown.

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